Question

A is known to speak truth $5$ out of $7$ times. What is the probability that $A$ reports that it is a $7$ when a die is thrown?

Answer 1

Here it is given that A speaks truth 5 out of 7 times. Here we should find the probability that A speaks lie since 7 cannot be thrown in a die. To find the probability of lying we subtract the probability of truth from the total probability.

Formula used: Let us consider, total outcomes of an event is $n$ and outcome of a particular event is $m$. So, the probability is $\dfrac{m}{n}.$

Complete step-by-step answer:
It is given that A is known to speak truth $5$ out of $7$ times. We have to find the probability that $A$ reports that it is a $7$ when a die is thrown.
Since 7 cannot be thrown in a die we should find the probability of A speaking lie.
We know that the total probability of an event is $1$.
If the total outcome of an event is $n$ and outcome of a particular event is $m$. The probability of getting the particular event is $\dfrac{m}{n}.$
Here we have $(m=5)$ & $(n=7)$
Therefore the probability of speaking the truth is $\dfrac{5}{7}$.
We can find the probability of lie by subtracting the probability of truth from the total probability,
Hence, the probability of lie $1 - \dfrac{5}{7} = \dfrac{2}{7}$
Hence, the probability that $A$ reports that it is a $7$ when a die is thrown is $\dfrac{2}{7}$ .

Additional information:
Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.
The probability of a certain event is $1$.

Note: We should be careful that while finding the probability that $A$ reports that it is a $7$ when a die is thrown because it should be chosen as a lie. Since it is given the probability of truth
We can also use the fact that if A is speaking truth $5$ out of $7$ times then he speaks lie 2 out of 7 times, and thereby using this fact the probability of lie can be found.